A185410 → D

nonnegative coefficients

Proved for all n all roots real, in (−∞, 0]coefficients of one sign Hadamard kernel interlacing (weak)

Recurrence

\(\left(-2 + n\right)\,P_{n}(x) = \left(3 - 12\,x + 24\,x^{2} - 3\,n + 16\,n x - 44\,n x^{2} - 4\,n^{2} x + 24\,n^{2} x^{2} - 4\,n^{3} x^{2}\right)\,P_{n-2}(x) + \left(2\,x - 10\,x^{2} + 8\,x^{3} - 2\,n x + 14\,n x^{2} - 12\,n x^{3} - 4\,n^{2} x^{2} + 4\,n^{2} x^{3}\right)\,P_{n-2}'(x) + \left(-7 + 16\,x + 4\,n - 16\,n x + 4\,n^{2} x\right)\,P_{n-1}(x) + \left(-4\,x + 4\,x^{2} + 2\,n x - 2\,n x^{2}\right)\,P_{n-1}'(x)\)

\(P_{0}(x) = 0\)

\(P_{1}(x) = 0\)

\(P_{2}(x) = 2\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	P_n(1)
0														0
1														0
2	2													2
3	10	8												18
4	36	120	24											180
5	116	1032	888	64										2100
6	358	7016	15504	5312	160									28350
7	1086	42240	192720	171840	28320	384								436590
8	3272	237264	1982640	3602240	1600320	140928	896							7567560
9	9832	1277936	18149232	58855360	53625920	13354368	671104	2048						145945800
10	29514	6704304	153881280	820085952	1325577120	688415616	103547136	3102720	4608					3101348250
11	88562	34584024	1238072256	10253224128	26882637600	24856328448	7978419456	762783744	14049792	10240				72020198250
12	265708	176462600	9600274440	118691225280	475140435840	710242251264	409666790400	85912872960	5415736320	62658560	22528			1814908995900
13	797148	894004152	72495799080	1298627614080	7600369299840	17215508588544	16088393373696	6142425016320	876114846720	37420738560	276215808	49152		49332526343100
14	2391470	4508283864	536972730576	13620869835520	112823183615040	369728702208000	522512540884992	325062048688128	85748463521280	8573523036160	253329838080	1206534144	106496	1438865351673750

Technical details

Node id: n-a1002d4e4f315530 · revision 0 · JSON
Definition source: derived:derivative
Fingerprint: f6-8ee1ad9f8b480ebb

Observed on the stored rows ?

Degrees: -1, -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12 (affine: slope 1, intercept -2)
Palindromic rows: no
γ-positive (gamma_positive): no
Nonnegative coefficients (nonn): yes
Only rational roots (rr_by_factors: every nonzero row is a product of linear factors over ℚ): no

Matches ?

No OEIS matches.

Submission text ?

Atlas rows n = 0, …, ascending powers of x

%e Example

Triangle begins:
  0:    0;
  1:    0;
  2:    2;
  3:   10,       8;
  4:   36,     120,       24;
  5:  116,    1032,      888,       64;
  6:  358,    7016,    15504,     5312,      160;
  7: 1086,   42240,   192720,   171840,    28320,      384;
  8: 3272,  237264,  1982640,  3602240,  1600320,   140928,    896;
  9: 9832, 1277936, 18149232, 58855360, 53625920, 13354368, 671104, 2048;

%F Formula

The row polynomials satisfy (n-2)*P(n,x) = -(3*n-3+(4*n^2-16*n+12)*x+(4*n^3-24*n^2+44*n-24)*x^2)*P(n-2,x) - ((2*n-2)*x+(4*n^2-14*n+10)*x^2-(4*n^2-12*n+8)*x^3)*d/dx P(n-2,x) + (4*n-7+(4*n^2-16*n+16)*x)*P(n-1,x) + ((2*n-4)*x-(2*n-4)*x^2)*d/dx P(n-1,x) for n >= 3, with P(0,x) = 0, P(1,x) = 0, P(2,x) = 2.

%t Mathematica

P[0, x_] := 0;
P[1, x_] := 0;
P[2, x_] := 2;
P[n_, x_] := P[n, x] = Expand[Together[(-(3*n - 3 + (4*n^2 - 16*n + 12)*x + (4*n^3 - 24*n^2 + 44*n - 24)*x^2)*P[n - 2, x] - ((2*n - 2)*x + (4*n^2 - 14*n + 10)*x^2 - (4*n^2 - 12*n + 8)*x^3)*D[P[n - 2, x], x] + (4*n - 7 + (4*n^2 - 16*n + 16)*x)*P[n - 1, x] + ((2*n - 4)*x - (2*n - 4)*x^2)*D[P[n - 1, x], x])/(n - 2)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N + 1)); v[1] = 0; v[2] = 0; v[3] = 2; for(n = 3, N, my(i = n + 1); v[i] = (-(3*n - 3 + (4*n^2 - 16*n + 12)*x + (4*n^3 - 24*n^2 + 44*n - 24)*x^2)*v[i - 2] - ((2*n - 2)*x + (4*n^2 - 14*n + 10)*x^2 - (4*n^2 - 12*n + 8)*x^3)*deriv(v[i - 2], x) + (4*n - 7 + (4*n^2 - 16*n + 16)*x)*v[i - 1] + ((2*n - 4)*x - (2*n - 4)*x^2)*deriv(v[i - 1], x))/(n - 2)); v};
row(n, p) = {my(c = Vecrev(p)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i - 1, v[i])))};
T(9)

polytool c48629374597

Properties

Real-rooted? Every row P_n, n ≥ 0, is zero or real-rooted with all roots in (−∞, 0]
by Laguerre’s theorem on differential operators
Coefficients of one sign? Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
derived from Laguerre’s theorem on differential operators
Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
derived from Laguerre’s theorem on differential operators
Interlacing of consecutive rows? Rows P_(m-1), P_m interlace (weak, unordered) for every m ≥ 1
by Laguerre’s theorem on differential operators

Proof chains ? JSON

Laguerre’s theorem on differential operators [SP14] polytool ? step ›
- from A decomposition of the double factorials A001147.?
  - Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ? step › direct check
  - Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
    
    from A triangular decomposition of the double factorial numbers A001147.?
    
    Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
    
    from A decomposition of the double factorials A001147.?
    
    ↑ Liu–Wang theorem (above)
    
    Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
    
    from A decomposition of the double factorials A001147.?
    
    ↑ Liu–Wang theorem (above)

References

[LW07] Lily L. Liu and Yi Wang. A unified approach to polynomial sequences with only real zeros. Advances in Applied Mathematics, 38(4):542–560, May 2007. doi:10.1016/j.aam.2006.02.003
[SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.